LMFDB - Embedded newform 1441.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1441,2,Mod(1,1441)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1441, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1441.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1441 = 11 \cdot 131 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1441.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$11.5064429313$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1441.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+2.00000 q^{2} +3.00000 q^{3} +2.00000 q^{4} -2.00000 q^{5} +6.00000 q^{6} +3.00000 q^{7} +6.00000 q^{9} +O(q^{10})$

\(q+2.00000 q^{2} +3.00000 q^{3} +2.00000 q^{4} -2.00000 q^{5} +6.00000 q^{6} +3.00000 q^{7} +6.00000 q^{9} -4.00000 q^{10} +1.00000 q^{11} +6.00000 q^{12} +1.00000 q^{13} +6.00000 q^{14} -6.00000 q^{15} -4.00000 q^{16} +2.00000 q^{17} +12.0000 q^{18} -4.00000 q^{19} -4.00000 q^{20} +9.00000 q^{21} +2.00000 q^{22} -1.00000 q^{25} +2.00000 q^{26} +9.00000 q^{27} +6.00000 q^{28} -12.0000 q^{30} -8.00000 q^{32} +3.00000 q^{33} +4.00000 q^{34} -6.00000 q^{35} +12.0000 q^{36} -2.00000 q^{37} -8.00000 q^{38} +3.00000 q^{39} +1.00000 q^{41} +18.0000 q^{42} -1.00000 q^{43} +2.00000 q^{44} -12.0000 q^{45} +8.00000 q^{47} -12.0000 q^{48} +2.00000 q^{49} -2.00000 q^{50} +6.00000 q^{51} +2.00000 q^{52} +3.00000 q^{53} +18.0000 q^{54} -2.00000 q^{55} -12.0000 q^{57} -3.00000 q^{59} -12.0000 q^{60} -15.0000 q^{61} +18.0000 q^{63} -8.00000 q^{64} -2.00000 q^{65} +6.00000 q^{66} +2.00000 q^{67} +4.00000 q^{68} -12.0000 q^{70} -10.0000 q^{71} -4.00000 q^{73} -4.00000 q^{74} -3.00000 q^{75} -8.00000 q^{76} +3.00000 q^{77} +6.00000 q^{78} +16.0000 q^{79} +8.00000 q^{80} +9.00000 q^{81} +2.00000 q^{82} +18.0000 q^{84} -4.00000 q^{85} -2.00000 q^{86} -7.00000 q^{89} -24.0000 q^{90} +3.00000 q^{91} +16.0000 q^{94} +8.00000 q^{95} -24.0000 q^{96} +14.0000 q^{97} +4.00000 q^{98} +6.00000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	2.00000	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$2$	2.00000	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$3$	3.00000	1.73205	0.866025	−	0.500000i	$-0.166667\pi$
$3$	3.00000	1.73205	0.866025	+	0.500000i	$0.166667\pi$
$4$	2.00000	1.00000
$5$	−2.00000	−0.894427	−0.447214	−	0.894427i	$-0.647584\pi$
$5$	−2.00000	−0.894427	−0.447214	+	0.894427i	$0.647584\pi$
$6$	6.00000	2.44949
$7$	3.00000	1.13389	0.566947	−	0.823754i	$-0.308125\pi$
$7$	3.00000	1.13389	0.566947	+	0.823754i	$0.308125\pi$
$8$	0	0
$9$	6.00000	2.00000
$10$	−4.00000	−1.26491
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	6.00000	1.73205
$13$	1.00000	0.277350	0.138675	−	0.990338i	$-0.455716\pi$
$13$	1.00000	0.277350	0.138675	+	0.990338i	$0.455716\pi$
$14$	6.00000	1.60357
$15$	−6.00000	−1.54919
$16$	−4.00000	−1.00000
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	12.0000	2.82843
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	−4.00000	−0.894427
$21$	9.00000	1.96396
$22$	2.00000	0.426401
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	2.00000	0.392232
$27$	9.00000	1.73205
$28$	6.00000	1.13389
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	−12.0000	−2.19089
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	−8.00000	−1.41421
$33$	3.00000	0.522233
$34$	4.00000	0.685994
$35$	−6.00000	−1.01419
$36$	12.0000	2.00000
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	−8.00000	−1.29777
$39$	3.00000	0.480384
$40$	0	0
$41$	1.00000	0.156174	0.0780869	−	0.996947i	$-0.475119\pi$
$41$	1.00000	0.156174	0.0780869	+	0.996947i	$0.475119\pi$
$42$	18.0000	2.77746
$43$	−1.00000	−0.152499	−0.0762493	−	0.997089i	$-0.524294\pi$
$43$	−1.00000	−0.152499	−0.0762493	+	0.997089i	$0.524294\pi$
$44$	2.00000	0.301511
$45$	−12.0000	−1.78885
$46$	0	0
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	−12.0000	−1.73205
$49$	2.00000	0.285714
$50$	−2.00000	−0.282843
$51$	6.00000	0.840168
$52$	2.00000	0.277350
$53$	3.00000	0.412082	0.206041	−	0.978543i	$-0.433942\pi$
$53$	3.00000	0.412082	0.206041	+	0.978543i	$0.433942\pi$
$54$	18.0000	2.44949
$55$	−2.00000	−0.269680
$56$	0	0
$57$	−12.0000	−1.58944
$58$	0	0
$59$	−3.00000	−0.390567	−0.195283	−	0.980747i	$-0.562563\pi$
$59$	−3.00000	−0.390567	−0.195283	+	0.980747i	$0.562563\pi$
$60$	−12.0000	−1.54919
$61$	−15.0000	−1.92055	−0.960277	−	0.279050i	$-0.909981\pi$
$61$	−15.0000	−1.92055	−0.960277	+	0.279050i	$0.909981\pi$
$62$	0	0
$63$	18.0000	2.26779
$64$	−8.00000	−1.00000
$65$	−2.00000	−0.248069
$66$	6.00000	0.738549
$67$	2.00000	0.244339	0.122169	−	0.992509i	$-0.461015\pi$
$67$	2.00000	0.244339	0.122169	+	0.992509i	$0.461015\pi$
$68$	4.00000	0.485071
$69$	0	0
$70$	−12.0000	−1.43427
$71$	−10.0000	−1.18678	−0.593391	−	0.804914i	$-0.702211\pi$
$71$	−10.0000	−1.18678	−0.593391	+	0.804914i	$0.702211\pi$
$72$	0	0
$73$	−4.00000	−0.468165	−0.234082	−	0.972217i	$-0.575209\pi$
$73$	−4.00000	−0.468165	−0.234082	+	0.972217i	$0.575209\pi$
$74$	−4.00000	−0.464991
$75$	−3.00000	−0.346410
$76$	−8.00000	−0.917663
$77$	3.00000	0.341882
$78$	6.00000	0.679366
$79$	16.0000	1.80014	0.900070	−	0.435745i	$-0.143515\pi$
$79$	16.0000	1.80014	0.900070	+	0.435745i	$0.143515\pi$
$80$	8.00000	0.894427
$81$	9.00000	1.00000
$82$	2.00000	0.220863
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	18.0000	1.96396
$85$	−4.00000	−0.433861
$86$	−2.00000	−0.215666
$87$	0	0
$88$	0	0
$89$	−7.00000	−0.741999	−0.370999	−	0.928633i	$-0.620985\pi$
$89$	−7.00000	−0.741999	−0.370999	+	0.928633i	$0.620985\pi$
$90$	−24.0000	−2.52982
$91$	3.00000	0.314485
$92$	0	0
$93$	0	0
$94$	16.0000	1.65027
$95$	8.00000	0.820783
$96$	−24.0000	−2.44949
$97$	14.0000	1.42148	0.710742	−	0.703452i	$-0.248359\pi$
$97$	14.0000	1.42148	0.710742	+	0.703452i	$0.248359\pi$
$98$	4.00000	0.404061
$99$	6.00000	0.603023
$100$	−2.00000	−0.200000
$101$	3.00000	0.298511	0.149256	−	0.988799i	$-0.452312\pi$
$101$	3.00000	0.298511	0.149256	+	0.988799i	$0.452312\pi$
$102$	12.0000	1.18818
$103$	6.00000	0.591198	0.295599	−	0.955312i	$-0.404481\pi$
$103$	6.00000	0.591198	0.295599	+	0.955312i	$0.404481\pi$
$104$	0	0
$105$	−18.0000	−1.75662
$106$	6.00000	0.582772
$107$	−15.0000	−1.45010	−0.725052	−	0.688694i	$-0.758184\pi$
$107$	−15.0000	−1.45010	−0.725052	+	0.688694i	$0.758184\pi$
$108$	18.0000	1.73205
$109$	1.00000	0.0957826	0.0478913	−	0.998853i	$-0.484750\pi$
$109$	1.00000	0.0957826	0.0478913	+	0.998853i	$0.484750\pi$
$110$	−4.00000	−0.381385
$111$	−6.00000	−0.569495
$112$	−12.0000	−1.13389
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	−24.0000	−2.24781
$115$	0	0
$116$	0	0
$117$	6.00000	0.554700
$118$	−6.00000	−0.552345
$119$	6.00000	0.550019
$120$	0	0
$121$	1.00000	0.0909091
$122$	−30.0000	−2.71607
$123$	3.00000	0.270501
$124$	0	0
$125$	12.0000	1.07331
$126$	36.0000	3.20713
$127$	4.00000	0.354943	0.177471	−	0.984126i	$-0.443208\pi$
$127$	4.00000	0.354943	0.177471	+	0.984126i	$0.443208\pi$
$128$	0	0
$129$	−3.00000	−0.264135
$130$	−4.00000	−0.350823
$131$	−1.00000	−0.0873704
$131$	−1.00000	−0.0873704
$132$	6.00000	0.522233
$133$	−12.0000	−1.04053
$134$	4.00000	0.345547
$135$	−18.0000	−1.54919
$136$	0	0
$137$	2.00000	0.170872	0.0854358	−	0.996344i	$-0.472772\pi$
$137$	2.00000	0.170872	0.0854358	+	0.996344i	$0.472772\pi$
$138$	0	0
$139$	−16.0000	−1.35710	−0.678551	−	0.734553i	$-0.737392\pi$
$139$	−16.0000	−1.35710	−0.678551	+	0.734553i	$0.737392\pi$
$140$	−12.0000	−1.01419
$141$	24.0000	2.02116
$142$	−20.0000	−1.67836
$143$	1.00000	0.0836242
$144$	−24.0000	−2.00000
$145$	0	0
$146$	−8.00000	−0.662085
$147$	6.00000	0.494872
$148$	−4.00000	−0.328798
$149$	−12.0000	−0.983078	−0.491539	−	0.870855i	$-0.663566\pi$
$149$	−12.0000	−0.983078	−0.491539	+	0.870855i	$0.663566\pi$
$150$	−6.00000	−0.489898
$151$	−17.0000	−1.38344	−0.691720	−	0.722166i	$-0.743147\pi$
$151$	−17.0000	−1.38344	−0.691720	+	0.722166i	$0.743147\pi$
$152$	0	0
$153$	12.0000	0.970143
$154$	6.00000	0.483494
$155$	0	0
$156$	6.00000	0.480384
$157$	−14.0000	−1.11732	−0.558661	−	0.829396i	$-0.688685\pi$
$157$	−14.0000	−1.11732	−0.558661	+	0.829396i	$0.688685\pi$
$158$	32.0000	2.54578
$159$	9.00000	0.713746
$160$	16.0000	1.26491
$161$	0	0
$162$	18.0000	1.41421
$163$	−14.0000	−1.09656	−0.548282	−	0.836293i	$-0.684718\pi$
$163$	−14.0000	−1.09656	−0.548282	+	0.836293i	$0.684718\pi$
$164$	2.00000	0.156174
$165$	−6.00000	−0.467099
$166$	0	0
$167$	16.0000	1.23812	0.619059	−	0.785345i	$-0.287514\pi$
$167$	16.0000	1.23812	0.619059	+	0.785345i	$0.287514\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	−8.00000	−0.613572
$171$	−24.0000	−1.83533
$172$	−2.00000	−0.152499
$173$	22.0000	1.67263	0.836315	−	0.548250i	$-0.184706\pi$
$173$	22.0000	1.67263	0.836315	+	0.548250i	$0.184706\pi$
$174$	0	0
$175$	−3.00000	−0.226779
$176$	−4.00000	−0.301511
$177$	−9.00000	−0.676481
$178$	−14.0000	−1.04934
$179$	5.00000	0.373718	0.186859	−	0.982387i	$-0.440169\pi$
$179$	5.00000	0.373718	0.186859	+	0.982387i	$0.440169\pi$
$180$	−24.0000	−1.78885
$181$	−18.0000	−1.33793	−0.668965	−	0.743294i	$-0.733262\pi$
$181$	−18.0000	−1.33793	−0.668965	+	0.743294i	$0.733262\pi$
$182$	6.00000	0.444750
$183$	−45.0000	−3.32650
$184$	0	0
$185$	4.00000	0.294086
$186$	0	0
$187$	2.00000	0.146254
$188$	16.0000	1.16692
$189$	27.0000	1.96396
$190$	16.0000	1.16076
$191$	9.00000	0.651217	0.325609	−	0.945505i	$-0.394431\pi$
$191$	9.00000	0.651217	0.325609	+	0.945505i	$0.394431\pi$
$192$	−24.0000	−1.73205
$193$	22.0000	1.58359	0.791797	−	0.610784i	$-0.209146\pi$
$193$	22.0000	1.58359	0.791797	+	0.610784i	$0.209146\pi$
$194$	28.0000	2.01028
$195$	−6.00000	−0.429669
$196$	4.00000	0.285714
$197$	18.0000	1.28245	0.641223	−	0.767354i	$-0.278427\pi$
$197$	18.0000	1.28245	0.641223	+	0.767354i	$0.278427\pi$
$198$	12.0000	0.852803
$199$	−8.00000	−0.567105	−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000	−0.567105	−0.283552	+	0.958957i	$0.591513\pi$
$200$	0	0
$201$	6.00000	0.423207
$202$	6.00000	0.422159
$203$	0	0
$204$	12.0000	0.840168
$205$	−2.00000	−0.139686
$206$	12.0000	0.836080
$207$	0	0
$208$	−4.00000	−0.277350
$209$	−4.00000	−0.276686
$210$	−36.0000	−2.48424
$211$	8.00000	0.550743	0.275371	−	0.961338i	$-0.411199\pi$
$211$	8.00000	0.550743	0.275371	+	0.961338i	$0.411199\pi$
$212$	6.00000	0.412082
$213$	−30.0000	−2.05557
$214$	−30.0000	−2.05076
$215$	2.00000	0.136399
$216$	0	0
$217$	0	0
$218$	2.00000	0.135457
$219$	−12.0000	−0.810885
$220$	−4.00000	−0.269680
$221$	2.00000	0.134535
$222$	−12.0000	−0.805387
$223$	−18.0000	−1.20537	−0.602685	−	0.797980i	$-0.705902\pi$
$223$	−18.0000	−1.20537	−0.602685	+	0.797980i	$0.705902\pi$
$224$	−24.0000	−1.60357
$225$	−6.00000	−0.400000
$226$	−12.0000	−0.798228
$227$	−18.0000	−1.19470	−0.597351	−	0.801980i	$-0.703780\pi$
$227$	−18.0000	−1.19470	−0.597351	+	0.801980i	$0.703780\pi$
$228$	−24.0000	−1.58944
$229$	20.0000	1.32164	0.660819	−	0.750546i	$-0.270209\pi$
$229$	20.0000	1.32164	0.660819	+	0.750546i	$0.270209\pi$
$230$	0	0
$231$	9.00000	0.592157
$232$	0	0
$233$	3.00000	0.196537	0.0982683	−	0.995160i	$-0.468670\pi$
$233$	3.00000	0.196537	0.0982683	+	0.995160i	$0.468670\pi$
$234$	12.0000	0.784465
$235$	−16.0000	−1.04372
$236$	−6.00000	−0.390567
$237$	48.0000	3.11794
$238$	12.0000	0.777844
$239$	12.0000	0.776215	0.388108	−	0.921614i	$-0.373129\pi$
$239$	12.0000	0.776215	0.388108	+	0.921614i	$0.373129\pi$
$240$	24.0000	1.54919
$241$	−12.0000	−0.772988	−0.386494	−	0.922292i	$-0.626314\pi$
$241$	−12.0000	−0.772988	−0.386494	+	0.922292i	$0.626314\pi$
$242$	2.00000	0.128565
$243$	0	0
$244$	−30.0000	−1.92055
$245$	−4.00000	−0.255551
$246$	6.00000	0.382546
$247$	−4.00000	−0.254514
$248$	0	0
$249$	0	0
$250$	24.0000	1.51789
$251$	−2.00000	−0.126239	−0.0631194	−	0.998006i	$-0.520105\pi$
$251$	−2.00000	−0.126239	−0.0631194	+	0.998006i	$0.520105\pi$
$252$	36.0000	2.26779
$253$	0	0
$254$	8.00000	0.501965
$255$	−12.0000	−0.751469
$256$	16.0000	1.00000
$257$	18.0000	1.12281	0.561405	−	0.827541i	$-0.310261\pi$
$257$	18.0000	1.12281	0.561405	+	0.827541i	$0.310261\pi$
$258$	−6.00000	−0.373544
$259$	−6.00000	−0.372822
$260$	−4.00000	−0.248069
$261$	0	0
$262$	−2.00000	−0.123560
$263$	21.0000	1.29492	0.647458	−	0.762101i	$-0.275832\pi$
$263$	21.0000	1.29492	0.647458	+	0.762101i	$0.275832\pi$
$264$	0	0
$265$	−6.00000	−0.368577
$266$	−24.0000	−1.47153
$267$	−21.0000	−1.28518
$268$	4.00000	0.244339
$269$	1.00000	0.0609711	0.0304855	−	0.999535i	$-0.490295\pi$
$269$	1.00000	0.0609711	0.0304855	+	0.999535i	$0.490295\pi$
$270$	−36.0000	−2.19089
$271$	9.00000	0.546711	0.273356	−	0.961913i	$-0.411866\pi$
$271$	9.00000	0.546711	0.273356	+	0.961913i	$0.411866\pi$
$272$	−8.00000	−0.485071
$273$	9.00000	0.544705
$274$	4.00000	0.241649
$275$	−1.00000	−0.0603023
$276$	0	0
$277$	22.0000	1.32185	0.660926	−	0.750451i	$-0.270164\pi$
$277$	22.0000	1.32185	0.660926	+	0.750451i	$0.270164\pi$
$278$	−32.0000	−1.91923
$279$	0	0
$280$	0	0
$281$	8.00000	0.477240	0.238620	−	0.971113i	$-0.423305\pi$
$281$	8.00000	0.477240	0.238620	+	0.971113i	$0.423305\pi$
$282$	48.0000	2.85836
$283$	13.0000	0.772770	0.386385	−	0.922338i	$-0.373724\pi$
$283$	13.0000	0.772770	0.386385	+	0.922338i	$0.373724\pi$
$284$	−20.0000	−1.18678
$285$	24.0000	1.42164
$286$	2.00000	0.118262
$287$	3.00000	0.177084
$288$	−48.0000	−2.82843
$289$	−13.0000	−0.764706
$290$	0	0
$291$	42.0000	2.46208
$292$	−8.00000	−0.468165
$293$	16.0000	0.934730	0.467365	−	0.884064i	$-0.345203\pi$
$293$	16.0000	0.934730	0.467365	+	0.884064i	$0.345203\pi$
$294$	12.0000	0.699854
$295$	6.00000	0.349334
$296$	0	0
$297$	9.00000	0.522233
$298$	−24.0000	−1.39028
$299$	0	0
$300$	−6.00000	−0.346410
$301$	−3.00000	−0.172917
$302$	−34.0000	−1.95648
$303$	9.00000	0.517036
$304$	16.0000	0.917663
$305$	30.0000	1.71780
$306$	24.0000	1.37199
$307$	−7.00000	−0.399511	−0.199756	−	0.979846i	$-0.564015\pi$
$307$	−7.00000	−0.399511	−0.199756	+	0.979846i	$0.564015\pi$
$308$	6.00000	0.341882
$309$	18.0000	1.02398
$310$	0	0
$311$	9.00000	0.510343	0.255172	−	0.966896i	$-0.417868\pi$
$311$	9.00000	0.510343	0.255172	+	0.966896i	$0.417868\pi$
$312$	0	0
$313$	12.0000	0.678280	0.339140	−	0.940736i	$-0.389864\pi$
$313$	12.0000	0.678280	0.339140	+	0.940736i	$0.389864\pi$
$314$	−28.0000	−1.58013
$315$	−36.0000	−2.02837
$316$	32.0000	1.80014
$317$	33.0000	1.85346	0.926732	−	0.375722i	$-0.122605\pi$
$317$	33.0000	1.85346	0.926732	+	0.375722i	$0.122605\pi$
$318$	18.0000	1.00939
$319$	0	0
$320$	16.0000	0.894427
$321$	−45.0000	−2.51166
$322$	0	0
$323$	−8.00000	−0.445132
$324$	18.0000	1.00000
$325$	−1.00000	−0.0554700
$326$	−28.0000	−1.55078
$327$	3.00000	0.165900
$328$	0	0
$329$	24.0000	1.32316
$330$	−12.0000	−0.660578
$331$	24.0000	1.31916	0.659580	−	0.751635i	$-0.270734\pi$
$331$	24.0000	1.31916	0.659580	+	0.751635i	$0.270734\pi$
$332$	0	0
$333$	−12.0000	−0.657596
$334$	32.0000	1.75096
$335$	−4.00000	−0.218543
$336$	−36.0000	−1.96396
$337$	−1.00000	−0.0544735	−0.0272367	−	0.999629i	$-0.508671\pi$
$337$	−1.00000	−0.0544735	−0.0272367	+	0.999629i	$0.508671\pi$
$338$	−24.0000	−1.30543
$339$	−18.0000	−0.977626
$340$	−8.00000	−0.433861
$341$	0	0
$342$	−48.0000	−2.59554
$343$	−15.0000	−0.809924
$344$	0	0
$345$	0	0
$346$	44.0000	2.36545
$347$	12.0000	0.644194	0.322097	−	0.946707i	$-0.395612\pi$
$347$	12.0000	0.644194	0.322097	+	0.946707i	$0.395612\pi$
$348$	0	0
$349$	24.0000	1.28469	0.642345	−	0.766415i	$-0.277962\pi$
$349$	24.0000	1.28469	0.642345	+	0.766415i	$0.277962\pi$
$350$	−6.00000	−0.320713
$351$	9.00000	0.480384
$352$	−8.00000	−0.426401
$353$	14.0000	0.745145	0.372572	−	0.928003i	$-0.378476\pi$
$353$	14.0000	0.745145	0.372572	+	0.928003i	$0.378476\pi$
$354$	−18.0000	−0.956689
$355$	20.0000	1.06149
$356$	−14.0000	−0.741999
$357$	18.0000	0.952661
$358$	10.0000	0.528516
$359$	18.0000	0.950004	0.475002	−	0.879985i	$-0.342447\pi$
$359$	18.0000	0.950004	0.475002	+	0.879985i	$0.342447\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	−36.0000	−1.89212
$363$	3.00000	0.157459
$364$	6.00000	0.314485
$365$	8.00000	0.418739
$366$	−90.0000	−4.70438
$367$	−27.0000	−1.40939	−0.704694	−	0.709511i	$-0.748916\pi$
$367$	−27.0000	−1.40939	−0.704694	+	0.709511i	$0.748916\pi$
$368$	0	0
$369$	6.00000	0.312348
$370$	8.00000	0.415900
$371$	9.00000	0.467257
$372$	0	0
$373$	−22.0000	−1.13912	−0.569558	−	0.821951i	$-0.692886\pi$
$373$	−22.0000	−1.13912	−0.569558	+	0.821951i	$0.692886\pi$
$374$	4.00000	0.206835
$375$	36.0000	1.85903
$376$	0	0
$377$	0	0
$378$	54.0000	2.77746
$379$	28.0000	1.43826	0.719132	−	0.694874i	$-0.244540\pi$
$379$	28.0000	1.43826	0.719132	+	0.694874i	$0.244540\pi$
$380$	16.0000	0.820783
$381$	12.0000	0.614779
$382$	18.0000	0.920960
$383$	15.0000	0.766464	0.383232	−	0.923652i	$-0.374811\pi$
$383$	15.0000	0.766464	0.383232	+	0.923652i	$0.374811\pi$
$384$	0	0
$385$	−6.00000	−0.305788
$386$	44.0000	2.23954
$387$	−6.00000	−0.304997
$388$	28.0000	1.42148
$389$	12.0000	0.608424	0.304212	−	0.952604i	$-0.401607\pi$
$389$	12.0000	0.608424	0.304212	+	0.952604i	$0.401607\pi$
$390$	−12.0000	−0.607644
$391$	0	0
$392$	0	0
$393$	−3.00000	−0.151330
$394$	36.0000	1.81365
$395$	−32.0000	−1.61009
$396$	12.0000	0.603023
$397$	3.00000	0.150566	0.0752828	−	0.997162i	$-0.476014\pi$
$397$	3.00000	0.150566	0.0752828	+	0.997162i	$0.476014\pi$
$398$	−16.0000	−0.802008
$399$	−36.0000	−1.80225
$400$	4.00000	0.200000
$401$	−14.0000	−0.699127	−0.349563	−	0.936913i	$-0.613670\pi$
$401$	−14.0000	−0.699127	−0.349563	+	0.936913i	$0.613670\pi$
$402$	12.0000	0.598506
$403$	0	0
$404$	6.00000	0.298511
$405$	−18.0000	−0.894427
$406$	0	0
$407$	−2.00000	−0.0991363
$408$	0	0
$409$	10.0000	0.494468	0.247234	−	0.968956i	$-0.420478\pi$
$409$	10.0000	0.494468	0.247234	+	0.968956i	$0.420478\pi$
$410$	−4.00000	−0.197546
$411$	6.00000	0.295958
$412$	12.0000	0.591198
$413$	−9.00000	−0.442861
$414$	0	0
$415$	0	0
$416$	−8.00000	−0.392232
$417$	−48.0000	−2.35057
$418$	−8.00000	−0.391293
$419$	−28.0000	−1.36789	−0.683945	−	0.729534i	$-0.739737\pi$
$419$	−28.0000	−1.36789	−0.683945	+	0.729534i	$0.739737\pi$
$420$	−36.0000	−1.75662
$421$	2.00000	0.0974740	0.0487370	−	0.998812i	$-0.484480\pi$
$421$	2.00000	0.0974740	0.0487370	+	0.998812i	$0.484480\pi$
$422$	16.0000	0.778868
$423$	48.0000	2.33384
$424$	0	0
$425$	−2.00000	−0.0970143
$426$	−60.0000	−2.90701
$427$	−45.0000	−2.17770
$428$	−30.0000	−1.45010
$429$	3.00000	0.144841
$430$	4.00000	0.192897
$431$	−16.0000	−0.770693	−0.385346	−	0.922772i	$-0.625918\pi$
$431$	−16.0000	−0.770693	−0.385346	+	0.922772i	$0.625918\pi$
$432$	−36.0000	−1.73205
$433$	10.0000	0.480569	0.240285	−	0.970702i	$-0.422759\pi$
$433$	10.0000	0.480569	0.240285	+	0.970702i	$0.422759\pi$
$434$	0	0
$435$	0	0
$436$	2.00000	0.0957826
$437$	0	0
$438$	−24.0000	−1.14676
$439$	−5.00000	−0.238637	−0.119318	−	0.992856i	$-0.538071\pi$
$439$	−5.00000	−0.238637	−0.119318	+	0.992856i	$0.538071\pi$
$440$	0	0
$441$	12.0000	0.571429
$442$	4.00000	0.190261
$443$	6.00000	0.285069	0.142534	−	0.989790i	$-0.454475\pi$
$443$	6.00000	0.285069	0.142534	+	0.989790i	$0.454475\pi$
$444$	−12.0000	−0.569495
$445$	14.0000	0.663664
$446$	−36.0000	−1.70465
$447$	−36.0000	−1.70274
$448$	−24.0000	−1.13389
$449$	18.0000	0.849473	0.424736	−	0.905317i	$-0.360367\pi$
$449$	18.0000	0.849473	0.424736	+	0.905317i	$0.360367\pi$
$450$	−12.0000	−0.565685
$451$	1.00000	0.0470882
$452$	−12.0000	−0.564433
$453$	−51.0000	−2.39619
$454$	−36.0000	−1.68956
$455$	−6.00000	−0.281284
$456$	0	0
$457$	5.00000	0.233890	0.116945	−	0.993138i	$-0.462690\pi$
$457$	5.00000	0.233890	0.116945	+	0.993138i	$0.462690\pi$
$458$	40.0000	1.86908
$459$	18.0000	0.840168
$460$	0	0
$461$	−42.0000	−1.95614	−0.978068	−	0.208288i	$-0.933211\pi$
$461$	−42.0000	−1.95614	−0.978068	+	0.208288i	$0.933211\pi$
$462$	18.0000	0.837436
$463$	−22.0000	−1.02243	−0.511213	−	0.859454i	$-0.670804\pi$
$463$	−22.0000	−1.02243	−0.511213	+	0.859454i	$0.670804\pi$
$464$	0	0
$465$	0	0
$466$	6.00000	0.277945
$467$	−32.0000	−1.48078	−0.740392	−	0.672176i	$-0.765360\pi$
$467$	−32.0000	−1.48078	−0.740392	+	0.672176i	$0.765360\pi$
$468$	12.0000	0.554700
$469$	6.00000	0.277054
$470$	−32.0000	−1.47605
$471$	−42.0000	−1.93526
$472$	0	0
$473$	−1.00000	−0.0459800
$474$	96.0000	4.40943
$475$	4.00000	0.183533
$476$	12.0000	0.550019
$477$	18.0000	0.824163
$478$	24.0000	1.09773
$479$	42.0000	1.91903	0.959514	−	0.281659i	$-0.0908848\pi$
$479$	42.0000	1.91903	0.959514	+	0.281659i	$0.0908848\pi$
$480$	48.0000	2.19089
$481$	−2.00000	−0.0911922
$482$	−24.0000	−1.09317
$483$	0	0
$484$	2.00000	0.0909091
$485$	−28.0000	−1.27141
$486$	0	0
$487$	−20.0000	−0.906287	−0.453143	−	0.891438i	$-0.649697\pi$
$487$	−20.0000	−0.906287	−0.453143	+	0.891438i	$0.649697\pi$
$488$	0	0
$489$	−42.0000	−1.89931
$490$	−8.00000	−0.361403
$491$	22.0000	0.992846	0.496423	−	0.868081i	$-0.334646\pi$
$491$	22.0000	0.992846	0.496423	+	0.868081i	$0.334646\pi$
$492$	6.00000	0.270501
$493$	0	0
$494$	−8.00000	−0.359937
$495$	−12.0000	−0.539360
$496$	0	0
$497$	−30.0000	−1.34568
$498$	0	0
$499$	2.00000	0.0895323	0.0447661	−	0.998997i	$-0.485746\pi$
$499$	2.00000	0.0895323	0.0447661	+	0.998997i	$0.485746\pi$
$500$	24.0000	1.07331
$501$	48.0000	2.14448
$502$	−4.00000	−0.178529
$503$	4.00000	0.178351	0.0891756	−	0.996016i	$-0.471577\pi$
$503$	4.00000	0.178351	0.0891756	+	0.996016i	$0.471577\pi$
$504$	0	0
$505$	−6.00000	−0.266996
$506$	0	0
$507$	−36.0000	−1.59882
$508$	8.00000	0.354943
$509$	−24.0000	−1.06378	−0.531891	−	0.846813i	$-0.678518\pi$
$509$	−24.0000	−1.06378	−0.531891	+	0.846813i	$0.678518\pi$
$510$	−24.0000	−1.06274
$511$	−12.0000	−0.530849
$512$	32.0000	1.41421
$513$	−36.0000	−1.58944
$514$	36.0000	1.58789
$515$	−12.0000	−0.528783
$516$	−6.00000	−0.264135
$517$	8.00000	0.351840
$518$	−12.0000	−0.527250
$519$	66.0000	2.89708
$520$	0	0
$521$	−4.00000	−0.175243	−0.0876216	−	0.996154i	$-0.527927\pi$
$521$	−4.00000	−0.175243	−0.0876216	+	0.996154i	$0.527927\pi$
$522$	0	0
$523$	14.0000	0.612177	0.306089	−	0.952003i	$-0.400980\pi$
$523$	14.0000	0.612177	0.306089	+	0.952003i	$0.400980\pi$
$524$	−2.00000	−0.0873704
$525$	−9.00000	−0.392792
$526$	42.0000	1.83129
$527$	0	0
$528$	−12.0000	−0.522233
$529$	−23.0000	−1.00000
$530$	−12.0000	−0.521247
$531$	−18.0000	−0.781133
$532$	−24.0000	−1.04053
$533$	1.00000	0.0433148
$534$	−42.0000	−1.81752
$535$	30.0000	1.29701
$536$	0	0
$537$	15.0000	0.647298
$538$	2.00000	0.0862261
$539$	2.00000	0.0861461
$540$	−36.0000	−1.54919
$541$	40.0000	1.71973	0.859867	−	0.510518i	$-0.170546\pi$
$541$	40.0000	1.71973	0.859867	+	0.510518i	$0.170546\pi$
$542$	18.0000	0.773166
$543$	−54.0000	−2.31736
$544$	−16.0000	−0.685994
$545$	−2.00000	−0.0856706
$546$	18.0000	0.770329
$547$	−26.0000	−1.11168	−0.555840	−	0.831289i	$-0.687603\pi$
$547$	−26.0000	−1.11168	−0.555840	+	0.831289i	$0.687603\pi$
$548$	4.00000	0.170872
$549$	−90.0000	−3.84111
$550$	−2.00000	−0.0852803
$551$	0	0
$552$	0	0
$553$	48.0000	2.04117
$554$	44.0000	1.86938
$555$	12.0000	0.509372
$556$	−32.0000	−1.35710
$557$	−3.00000	−0.127114	−0.0635570	−	0.997978i	$-0.520244\pi$
$557$	−3.00000	−0.127114	−0.0635570	+	0.997978i	$0.520244\pi$
$558$	0	0
$559$	−1.00000	−0.0422955
$560$	24.0000	1.01419
$561$	6.00000	0.253320
$562$	16.0000	0.674919
$563$	−11.0000	−0.463595	−0.231797	−	0.972764i	$-0.574461\pi$
$563$	−11.0000	−0.463595	−0.231797	+	0.972764i	$0.574461\pi$
$564$	48.0000	2.02116
$565$	12.0000	0.504844
$566$	26.0000	1.09286
$567$	27.0000	1.13389
$568$	0	0
$569$	13.0000	0.544988	0.272494	−	0.962157i	$-0.412151\pi$
$569$	13.0000	0.544988	0.272494	+	0.962157i	$0.412151\pi$
$570$	48.0000	2.01050
$571$	−16.0000	−0.669579	−0.334790	−	0.942293i	$-0.608665\pi$
$571$	−16.0000	−0.669579	−0.334790	+	0.942293i	$0.608665\pi$
$572$	2.00000	0.0836242
$573$	27.0000	1.12794
$574$	6.00000	0.250435
$575$	0	0
$576$	−48.0000	−2.00000
$577$	−7.00000	−0.291414	−0.145707	−	0.989328i	$-0.546546\pi$
$577$	−7.00000	−0.291414	−0.145707	+	0.989328i	$0.546546\pi$
$578$	−26.0000	−1.08146
$579$	66.0000	2.74287
$580$	0	0
$581$	0	0
$582$	84.0000	3.48191
$583$	3.00000	0.124247
$584$	0	0
$585$	−12.0000	−0.496139
$586$	32.0000	1.32191
$587$	7.00000	0.288921	0.144460	−	0.989511i	$-0.453855\pi$
$587$	7.00000	0.288921	0.144460	+	0.989511i	$0.453855\pi$
$588$	12.0000	0.494872
$589$	0	0
$590$	12.0000	0.494032
$591$	54.0000	2.22126
$592$	8.00000	0.328798
$593$	−6.00000	−0.246390	−0.123195	−	0.992382i	$-0.539314\pi$
$593$	−6.00000	−0.246390	−0.123195	+	0.992382i	$0.539314\pi$
$594$	18.0000	0.738549
$595$	−12.0000	−0.491952
$596$	−24.0000	−0.983078
$597$	−24.0000	−0.982255
$598$	0	0
$599$	15.0000	0.612883	0.306442	−	0.951889i	$-0.400862\pi$
$599$	15.0000	0.612883	0.306442	+	0.951889i	$0.400862\pi$
$600$	0	0
$601$	21.0000	0.856608	0.428304	−	0.903635i	$-0.359111\pi$
$601$	21.0000	0.856608	0.428304	+	0.903635i	$0.359111\pi$
$602$	−6.00000	−0.244542
$603$	12.0000	0.488678
$604$	−34.0000	−1.38344
$605$	−2.00000	−0.0813116
$606$	18.0000	0.731200
$607$	34.0000	1.38002	0.690009	−	0.723801i	$-0.257607\pi$
$607$	34.0000	1.38002	0.690009	+	0.723801i	$0.257607\pi$
$608$	32.0000	1.29777
$609$	0	0
$610$	60.0000	2.42933
$611$	8.00000	0.323645
$612$	24.0000	0.970143
$613$	14.0000	0.565455	0.282727	−	0.959200i	$-0.408761\pi$
$613$	14.0000	0.565455	0.282727	+	0.959200i	$0.408761\pi$
$614$	−14.0000	−0.564994
$615$	−6.00000	−0.241943
$616$	0	0
$617$	10.0000	0.402585	0.201292	−	0.979531i	$-0.435486\pi$
$617$	10.0000	0.402585	0.201292	+	0.979531i	$0.435486\pi$
$618$	36.0000	1.44813
$619$	−44.0000	−1.76851	−0.884255	−	0.467005i	$-0.845333\pi$
$619$	−44.0000	−1.76851	−0.884255	+	0.467005i	$0.845333\pi$
$620$	0	0
$621$	0	0
$622$	18.0000	0.721734
$623$	−21.0000	−0.841347
$624$	−12.0000	−0.480384
$625$	−19.0000	−0.760000
$626$	24.0000	0.959233
$627$	−12.0000	−0.479234
$628$	−28.0000	−1.11732
$629$	−4.00000	−0.159490
$630$	−72.0000	−2.86855
$631$	−39.0000	−1.55257	−0.776283	−	0.630385i	$-0.782897\pi$
$631$	−39.0000	−1.55257	−0.776283	+	0.630385i	$0.782897\pi$
$632$	0	0
$633$	24.0000	0.953914
$634$	66.0000	2.62119
$635$	−8.00000	−0.317470
$636$	18.0000	0.713746
$637$	2.00000	0.0792429
$638$	0	0
$639$	−60.0000	−2.37356
$640$	0	0
$641$	5.00000	0.197488	0.0987441	−	0.995113i	$-0.468517\pi$
$641$	5.00000	0.197488	0.0987441	+	0.995113i	$0.468517\pi$
$642$	−90.0000	−3.55202
$643$	26.0000	1.02534	0.512670	−	0.858586i	$-0.328656\pi$
$643$	26.0000	1.02534	0.512670	+	0.858586i	$0.328656\pi$
$644$	0	0
$645$	6.00000	0.236250
$646$	−16.0000	−0.629512
$647$	12.0000	0.471769	0.235884	−	0.971781i	$-0.424201\pi$
$647$	12.0000	0.471769	0.235884	+	0.971781i	$0.424201\pi$
$648$	0	0
$649$	−3.00000	−0.117760
$650$	−2.00000	−0.0784465
$651$	0	0
$652$	−28.0000	−1.09656
$653$	5.00000	0.195665	0.0978326	−	0.995203i	$-0.468809\pi$
$653$	5.00000	0.195665	0.0978326	+	0.995203i	$0.468809\pi$
$654$	6.00000	0.234619
$655$	2.00000	0.0781465
$656$	−4.00000	−0.156174
$657$	−24.0000	−0.936329
$658$	48.0000	1.87123
$659$	−20.0000	−0.779089	−0.389545	−	0.921008i	$-0.627368\pi$
$659$	−20.0000	−0.779089	−0.389545	+	0.921008i	$0.627368\pi$
$660$	−12.0000	−0.467099
$661$	−38.0000	−1.47803	−0.739014	−	0.673690i	$-0.764708\pi$
$661$	−38.0000	−1.47803	−0.739014	+	0.673690i	$0.764708\pi$
$662$	48.0000	1.86557
$663$	6.00000	0.233021
$664$	0	0
$665$	24.0000	0.930680
$666$	−24.0000	−0.929981
$667$	0	0
$668$	32.0000	1.23812
$669$	−54.0000	−2.08776
$670$	−8.00000	−0.309067
$671$	−15.0000	−0.579069
$672$	−72.0000	−2.77746
$673$	6.00000	0.231283	0.115642	−	0.993291i	$-0.463108\pi$
$673$	6.00000	0.231283	0.115642	+	0.993291i	$0.463108\pi$
$674$	−2.00000	−0.0770371
$675$	−9.00000	−0.346410
$676$	−24.0000	−0.923077
$677$	20.0000	0.768662	0.384331	−	0.923195i	$-0.374432\pi$
$677$	20.0000	0.768662	0.384331	+	0.923195i	$0.374432\pi$
$678$	−36.0000	−1.38257
$679$	42.0000	1.61181
$680$	0	0
$681$	−54.0000	−2.06928
$682$	0	0
$683$	3.00000	0.114792	0.0573959	−	0.998351i	$-0.481720\pi$
$683$	3.00000	0.114792	0.0573959	+	0.998351i	$0.481720\pi$
$684$	−48.0000	−1.83533
$685$	−4.00000	−0.152832
$686$	−30.0000	−1.14541
$687$	60.0000	2.28914
$688$	4.00000	0.152499
$689$	3.00000	0.114291
$690$	0	0
$691$	−43.0000	−1.63580	−0.817899	−	0.575362i	$-0.804861\pi$
$691$	−43.0000	−1.63580	−0.817899	+	0.575362i	$0.804861\pi$
$692$	44.0000	1.67263
$693$	18.0000	0.683763
$694$	24.0000	0.911028
$695$	32.0000	1.21383
$696$	0	0
$697$	2.00000	0.0757554
$698$	48.0000	1.81683
$699$	9.00000	0.340411
$700$	−6.00000	−0.226779
$701$	−5.00000	−0.188847	−0.0944237	−	0.995532i	$-0.530101\pi$
$701$	−5.00000	−0.188847	−0.0944237	+	0.995532i	$0.530101\pi$
$702$	18.0000	0.679366
$703$	8.00000	0.301726
$704$	−8.00000	−0.301511
$705$	−48.0000	−1.80778
$706$	28.0000	1.05379
$707$	9.00000	0.338480
$708$	−18.0000	−0.676481
$709$	10.0000	0.375558	0.187779	−	0.982211i	$-0.439871\pi$
$709$	10.0000	0.375558	0.187779	+	0.982211i	$0.439871\pi$
$710$	40.0000	1.50117
$711$	96.0000	3.60028
$712$	0	0
$713$	0	0
$714$	36.0000	1.34727
$715$	−2.00000	−0.0747958
$716$	10.0000	0.373718
$717$	36.0000	1.34444
$718$	36.0000	1.34351
$719$	−5.00000	−0.186469	−0.0932343	−	0.995644i	$-0.529721\pi$
$719$	−5.00000	−0.186469	−0.0932343	+	0.995644i	$0.529721\pi$
$720$	48.0000	1.78885
$721$	18.0000	0.670355
$722$	−6.00000	−0.223297
$723$	−36.0000	−1.33885
$724$	−36.0000	−1.33793
$725$	0	0
$726$	6.00000	0.222681
$727$	−40.0000	−1.48352	−0.741759	−	0.670667i	$-0.766008\pi$
$727$	−40.0000	−1.48352	−0.741759	+	0.670667i	$0.766008\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	16.0000	0.592187
$731$	−2.00000	−0.0739727
$732$	−90.0000	−3.32650
$733$	36.0000	1.32969	0.664845	−	0.746981i	$-0.268498\pi$
$733$	36.0000	1.32969	0.664845	+	0.746981i	$0.268498\pi$
$734$	−54.0000	−1.99318
$735$	−12.0000	−0.442627
$736$	0	0
$737$	2.00000	0.0736709
$738$	12.0000	0.441726
$739$	−45.0000	−1.65535	−0.827676	−	0.561206i	$-0.810337\pi$
$739$	−45.0000	−1.65535	−0.827676	+	0.561206i	$0.810337\pi$
$740$	8.00000	0.294086
$741$	−12.0000	−0.440831
$742$	18.0000	0.660801
$743$	48.0000	1.76095	0.880475	−	0.474093i	$-0.157224\pi$
$743$	48.0000	1.76095	0.880475	+	0.474093i	$0.157224\pi$
$744$	0	0
$745$	24.0000	0.879292
$746$	−44.0000	−1.61095
$747$	0	0
$748$	4.00000	0.146254
$749$	−45.0000	−1.64426
$750$	72.0000	2.62907
$751$	22.0000	0.802791	0.401396	−	0.915905i	$-0.368525\pi$
$751$	22.0000	0.802791	0.401396	+	0.915905i	$0.368525\pi$
$752$	−32.0000	−1.16692
$753$	−6.00000	−0.218652
$754$	0	0
$755$	34.0000	1.23739
$756$	54.0000	1.96396
$757$	−23.0000	−0.835949	−0.417975	−	0.908459i	$-0.637260\pi$
$757$	−23.0000	−0.835949	−0.417975	+	0.908459i	$0.637260\pi$
$758$	56.0000	2.03401
$759$	0	0
$760$	0	0
$761$	−38.0000	−1.37750	−0.688749	−	0.724999i	$-0.741840\pi$
$761$	−38.0000	−1.37750	−0.688749	+	0.724999i	$0.741840\pi$
$762$	24.0000	0.869428
$763$	3.00000	0.108607
$764$	18.0000	0.651217
$765$	−24.0000	−0.867722
$766$	30.0000	1.08394
$767$	−3.00000	−0.108324
$768$	48.0000	1.73205
$769$	14.0000	0.504853	0.252426	−	0.967616i	$-0.418771\pi$
$769$	14.0000	0.504853	0.252426	+	0.967616i	$0.418771\pi$
$770$	−12.0000	−0.432450
$771$	54.0000	1.94476
$772$	44.0000	1.58359
$773$	22.0000	0.791285	0.395643	−	0.918405i	$-0.370522\pi$
$773$	22.0000	0.791285	0.395643	+	0.918405i	$0.370522\pi$
$774$	−12.0000	−0.431331
$775$	0	0
$776$	0	0
$777$	−18.0000	−0.645746
$778$	24.0000	0.860442
$779$	−4.00000	−0.143315
$780$	−12.0000	−0.429669
$781$	−10.0000	−0.357828
$782$	0	0
$783$	0	0
$784$	−8.00000	−0.285714
$785$	28.0000	0.999363
$786$	−6.00000	−0.214013
$787$	−4.00000	−0.142585	−0.0712923	−	0.997455i	$-0.522712\pi$
$787$	−4.00000	−0.142585	−0.0712923	+	0.997455i	$0.522712\pi$
$788$	36.0000	1.28245
$789$	63.0000	2.24286
$790$	−64.0000	−2.27702
$791$	−18.0000	−0.640006
$792$	0	0
$793$	−15.0000	−0.532666
$794$	6.00000	0.212932
$795$	−18.0000	−0.638394
$796$	−16.0000	−0.567105
$797$	−49.0000	−1.73567	−0.867835	−	0.496853i	$-0.834489\pi$
$797$	−49.0000	−1.73567	−0.867835	+	0.496853i	$0.834489\pi$
$798$	−72.0000	−2.54877
$799$	16.0000	0.566039
$800$	8.00000	0.282843
$801$	−42.0000	−1.48400
$802$	−28.0000	−0.988714
$803$	−4.00000	−0.141157
$804$	12.0000	0.423207
$805$	0	0
$806$	0	0
$807$	3.00000	0.105605
$808$	0	0
$809$	2.00000	0.0703163	0.0351581	−	0.999382i	$-0.488807\pi$
$809$	2.00000	0.0703163	0.0351581	+	0.999382i	$0.488807\pi$
$810$	−36.0000	−1.26491
$811$	31.0000	1.08856	0.544279	−	0.838905i	$-0.316803\pi$
$811$	31.0000	1.08856	0.544279	+	0.838905i	$0.316803\pi$
$812$	0	0
$813$	27.0000	0.946931
$814$	−4.00000	−0.140200
$815$	28.0000	0.980797
$816$	−24.0000	−0.840168
$817$	4.00000	0.139942
$818$	20.0000	0.699284
$819$	18.0000	0.628971
$820$	−4.00000	−0.139686
$821$	−39.0000	−1.36111	−0.680555	−	0.732697i	$-0.738261\pi$
$821$	−39.0000	−1.36111	−0.680555	+	0.732697i	$0.738261\pi$
$822$	12.0000	0.418548
$823$	16.0000	0.557725	0.278862	−	0.960331i	$-0.410043\pi$
$823$	16.0000	0.557725	0.278862	+	0.960331i	$0.410043\pi$
$824$	0	0
$825$	−3.00000	−0.104447
$826$	−18.0000	−0.626300
$827$	−7.00000	−0.243414	−0.121707	−	0.992566i	$-0.538837\pi$
$827$	−7.00000	−0.243414	−0.121707	+	0.992566i	$0.538837\pi$
$828$	0	0
$829$	−21.0000	−0.729360	−0.364680	−	0.931133i	$-0.618822\pi$
$829$	−21.0000	−0.729360	−0.364680	+	0.931133i	$0.618822\pi$
$830$	0	0
$831$	66.0000	2.28951
$832$	−8.00000	−0.277350
$833$	4.00000	0.138592
$834$	−96.0000	−3.32421
$835$	−32.0000	−1.10741
$836$	−8.00000	−0.276686
$837$	0	0
$838$	−56.0000	−1.93449
$839$	15.0000	0.517858	0.258929	−	0.965896i	$-0.416631\pi$
$839$	15.0000	0.517858	0.258929	+	0.965896i	$0.416631\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	4.00000	0.137849
$843$	24.0000	0.826604
$844$	16.0000	0.550743
$845$	24.0000	0.825625
$846$	96.0000	3.30055
$847$	3.00000	0.103081
$848$	−12.0000	−0.412082
$849$	39.0000	1.33848
$850$	−4.00000	−0.137199
$851$	0	0
$852$	−60.0000	−2.05557
$853$	26.0000	0.890223	0.445112	−	0.895475i	$-0.353164\pi$
$853$	26.0000	0.890223	0.445112	+	0.895475i	$0.353164\pi$
$854$	−90.0000	−3.07974
$855$	48.0000	1.64157
$856$	0	0
$857$	12.0000	0.409912	0.204956	−	0.978771i	$-0.434295\pi$
$857$	12.0000	0.409912	0.204956	+	0.978771i	$0.434295\pi$
$858$	6.00000	0.204837
$859$	20.0000	0.682391	0.341196	−	0.939992i	$-0.389168\pi$
$859$	20.0000	0.682391	0.341196	+	0.939992i	$0.389168\pi$
$860$	4.00000	0.136399
$861$	9.00000	0.306719
$862$	−32.0000	−1.08992
$863$	44.0000	1.49778	0.748889	−	0.662696i	$-0.230588\pi$
$863$	44.0000	1.49778	0.748889	+	0.662696i	$0.230588\pi$
$864$	−72.0000	−2.44949
$865$	−44.0000	−1.49604
$866$	20.0000	0.679628
$867$	−39.0000	−1.32451
$868$	0	0
$869$	16.0000	0.542763
$870$	0	0
$871$	2.00000	0.0677674
$872$	0	0
$873$	84.0000	2.84297
$874$	0	0
$875$	36.0000	1.21702
$876$	−24.0000	−0.810885
$877$	−27.0000	−0.911725	−0.455863	−	0.890050i	$-0.650669\pi$
$877$	−27.0000	−0.911725	−0.455863	+	0.890050i	$0.650669\pi$
$878$	−10.0000	−0.337484
$879$	48.0000	1.61900
$880$	8.00000	0.269680
$881$	2.00000	0.0673817	0.0336909	−	0.999432i	$-0.489274\pi$
$881$	2.00000	0.0673817	0.0336909	+	0.999432i	$0.489274\pi$
$882$	24.0000	0.808122
$883$	16.0000	0.538443	0.269221	−	0.963078i	$-0.413234\pi$
$883$	16.0000	0.538443	0.269221	+	0.963078i	$0.413234\pi$
$884$	4.00000	0.134535
$885$	18.0000	0.605063
$886$	12.0000	0.403148
$887$	28.0000	0.940148	0.470074	−	0.882627i	$-0.344227\pi$
$887$	28.0000	0.940148	0.470074	+	0.882627i	$0.344227\pi$
$888$	0	0
$889$	12.0000	0.402467
$890$	28.0000	0.938562
$891$	9.00000	0.301511
$892$	−36.0000	−1.20537
$893$	−32.0000	−1.07084
$894$	−72.0000	−2.40804
$895$	−10.0000	−0.334263
$896$	0	0
$897$	0	0
$898$	36.0000	1.20134
$899$	0	0
$900$	−12.0000	−0.400000
$901$	6.00000	0.199889
$902$	2.00000	0.0665927
$903$	−9.00000	−0.299501
$904$	0	0
$905$	36.0000	1.19668
$906$	−102.000	−3.38872
$907$	13.0000	0.431658	0.215829	−	0.976431i	$-0.430755\pi$
$907$	13.0000	0.431658	0.215829	+	0.976431i	$0.430755\pi$
$908$	−36.0000	−1.19470
$909$	18.0000	0.597022
$910$	−12.0000	−0.397796
$911$	−45.0000	−1.49092	−0.745458	−	0.666552i	$-0.767769\pi$
$911$	−45.0000	−1.49092	−0.745458	+	0.666552i	$0.767769\pi$
$912$	48.0000	1.58944
$913$	0	0
$914$	10.0000	0.330771
$915$	90.0000	2.97531
$916$	40.0000	1.32164
$917$	−3.00000	−0.0990687
$918$	36.0000	1.18818
$919$	38.0000	1.25350	0.626752	−	0.779219i	$-0.284384\pi$
$919$	38.0000	1.25350	0.626752	+	0.779219i	$0.284384\pi$
$920$	0	0
$921$	−21.0000	−0.691974
$922$	−84.0000	−2.76639
$923$	−10.0000	−0.329154
$924$	18.0000	0.592157
$925$	2.00000	0.0657596
$926$	−44.0000	−1.44593
$927$	36.0000	1.18240
$928$	0	0
$929$	−21.0000	−0.688988	−0.344494	−	0.938789i	$-0.611949\pi$
$929$	−21.0000	−0.688988	−0.344494	+	0.938789i	$0.611949\pi$
$930$	0	0
$931$	−8.00000	−0.262189
$932$	6.00000	0.196537
$933$	27.0000	0.883940
$934$	−64.0000	−2.09414
$935$	−4.00000	−0.130814
$936$	0	0
$937$	−7.00000	−0.228680	−0.114340	−	0.993442i	$-0.536475\pi$
$937$	−7.00000	−0.228680	−0.114340	+	0.993442i	$0.536475\pi$
$938$	12.0000	0.391814
$939$	36.0000	1.17482
$940$	−32.0000	−1.04372
$941$	−18.0000	−0.586783	−0.293392	−	0.955992i	$-0.594784\pi$
$941$	−18.0000	−0.586783	−0.293392	+	0.955992i	$0.594784\pi$
$942$	−84.0000	−2.73687
$943$	0	0
$944$	12.0000	0.390567
$945$	−54.0000	−1.75662
$946$	−2.00000	−0.0650256
$947$	18.0000	0.584921	0.292461	−	0.956278i	$-0.405526\pi$
$947$	18.0000	0.584921	0.292461	+	0.956278i	$0.405526\pi$
$948$	96.0000	3.11794
$949$	−4.00000	−0.129845
$950$	8.00000	0.259554
$951$	99.0000	3.21029
$952$	0	0
$953$	−14.0000	−0.453504	−0.226752	−	0.973952i	$-0.572811\pi$
$953$	−14.0000	−0.453504	−0.226752	+	0.973952i	$0.572811\pi$
$954$	36.0000	1.16554
$955$	−18.0000	−0.582466
$956$	24.0000	0.776215
$957$	0	0
$958$	84.0000	2.71392
$959$	6.00000	0.193750
$960$	48.0000	1.54919
$961$	−31.0000	−1.00000
$962$	−4.00000	−0.128965
$963$	−90.0000	−2.90021
$964$	−24.0000	−0.772988
$965$	−44.0000	−1.41641
$966$	0	0
$967$	24.0000	0.771788	0.385894	−	0.922543i	$-0.373893\pi$
$967$	24.0000	0.771788	0.385894	+	0.922543i	$0.373893\pi$
$968$	0	0
$969$	−24.0000	−0.770991
$970$	−56.0000	−1.79805
$971$	−38.0000	−1.21948	−0.609739	−	0.792602i	$-0.708726\pi$
$971$	−38.0000	−1.21948	−0.609739	+	0.792602i	$0.708726\pi$
$972$	0	0
$973$	−48.0000	−1.53881
$974$	−40.0000	−1.28168
$975$	−3.00000	−0.0960769
$976$	60.0000	1.92055
$977$	6.00000	0.191957	0.0959785	−	0.995383i	$-0.469402\pi$
$977$	6.00000	0.191957	0.0959785	+	0.995383i	$0.469402\pi$
$978$	−84.0000	−2.68602
$979$	−7.00000	−0.223721
$980$	−8.00000	−0.255551
$981$	6.00000	0.191565
$982$	44.0000	1.40410
$983$	−22.0000	−0.701691	−0.350846	−	0.936433i	$-0.614106\pi$
$983$	−22.0000	−0.701691	−0.350846	+	0.936433i	$0.614106\pi$
$984$	0	0
$985$	−36.0000	−1.14706
$986$	0	0
$987$	72.0000	2.29179
$988$	−8.00000	−0.254514
$989$	0	0
$990$	−24.0000	−0.762770
$991$	−61.0000	−1.93773	−0.968864	−	0.247592i	$-0.920361\pi$
$991$	−61.0000	−1.93773	−0.968864	+	0.247592i	$0.920361\pi$
$992$	0	0
$993$	72.0000	2.28485
$994$	−60.0000	−1.90308
$995$	16.0000	0.507234
$996$	0	0
$997$	−33.0000	−1.04512	−0.522560	−	0.852602i	$-0.675023\pi$
$997$	−33.0000	−1.04512	−0.522560	+	0.852602i	$0.675023\pi$
$998$	4.00000	0.126618
$999$	−18.0000	−0.569495

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1441.2.a.b.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1441.2.a.b.1.1	✓	1	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1441 = 11 \cdot 131 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1441.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more